Why can a Venn layout for 4+ establishes not be created making use of circles?

This page offers a couple of instances of Venn layouts for 4 collections. Some instances:

Thinking of it for a little, it is difficult to dividing the aircraft right into the $16$ sectors needed for a full $4$- set Venn layout making use of just circles as we can provide for $<4$ collections. Yet it is practical with ellipses or rectangular shapes, so we do not call for non-convex forms as Edwards usages.

So what buildings of a form establish its viability for $n$- set Venn layouts? Especially, why are circles unsatisfactory for the instance $n=4$?

2019-05-05 00:52:08
Source Share
Answers: 1

The brief solution, from a paper by Frank Ruskey, Carla D. Savage, and also Stan Wagon is as adheres to :

... it is difficult to attract a Venn layout with circles that will certainly stand for all the feasible junctions of 4 (or even more) collections. This is a straightforward effect of the reality that circles can finitely converge in at the majority of 2 factors and also Euler’s relation F − E+V = 2 for the variety of faces, sides, and also vertices in an aircraft chart.

The very same paper takes place in fairly some information concerning the procedure of developing Venn layouts for greater values of n , specifically for straightforward layouts with rotational proportion.

For a straightforward recap, the most effective solution I can locate got on WikiAnswers :

Two circles converge in at the majority of 2 factors, and also each junction develops one new area. (Going clockwise around the circle, the contour from each junction to the next separates a present area right into 2.)

Given that the 4th circle converges the first 3 in at the majority of 6 areas, it develops at the majority of 6 new areas ; that's 14 complete, yet you require 2 ^ 4 = 16 areas to stand for all feasible partnerships in between 4 collections.

Yet you can create a Venn layout for 4 collections with 4 ellipses, due to the fact that 2 ellipses can converge in greater than 2 factors.

Both of these resources show that the essential building of a form that would certainly make it ideal or improper for greater - order Venn layouts is the variety of feasible junctions (and also consequently, below - areas) that can be used 2 of the very same form.

To highlight better, take into consideration several of the intricate forms made use of for n = 5, n = 7 and also n = 11 (from Wolfram Mathworld) :

The framework of these forms is picked such that they can converge with each - various other in as several means as called for to generate the variety of one-of-a-kind areas needed for an offered n .

See additionally : Are Venn Diagrams Limited to Three or Fewer Sets?

2019-05-08 19:22:02